MATROIDS ON DOMAINS OF FUNCTIONS

BY

KEVIN .PRENDERGAST B.E.(Melb.)B.A.(Melb.)

B.A.(Hons.)(Tas.)

Submitted in fulfilment of the

requirements for the degree of

Master of Science

UNIVERSITY OF TASMANIA

HOBART

1980

Except as stated herein, this thesis contains no material

which has been accepted for the award of any other degree or diploma

in any university, and to the best of the author's knowledge

and belief, contains no copy or paraphrase of material previously

published or written by another person, except when due reference

is made in the text of the thesis.

K. Prenderga .

The author gratefully acknowledges the inspiration and advice

of his supervisor, Don Row, and the many pleasant hours of

mathematics spent with him.

This thesis is for Wendy, Katrina, Lisa, Jacinth and Natasha,

who by their encouragement made it possible.

CONTENTS

SUMMARY

INTRODUCTION vi

NOTATION

CHAPTER 1 1

CHAPTER 2 7

CHAPTER 3 25

CHAPTER 4 32

CHAPTER 5 50

CHAPTER 6 54

CHAPTER 7 59

BIBLIOGRAPHY 75

INDEX 77

iv

SUMMARY

Any integer-valued function with finite domain E defines,

by means of an associated submodular function on 2E

, a matroid

M(E).

The class pi of matroids so obtained is closed under restriction,

contraction, and is self dual. We show it consists precisely of those

transversal matroids having a presentation in which the sets of the

presentation are nested.

We give an excluded minor characterisation of M .

We count the members of M on an n-set and exhibit explicitly

those on a 6-set.

We extend the above investigation, using Rado's Selection

Principle, and permitting E to be infinite, to pregeometries.

Finally, by examining some integer-valued functions on E r

with r possibly greater than 1, we discuss some of the properties

of the class of matroids so obtained.

vi

INTRODUCTION

A matroid is essentially a set with an independence structure

. defined on its subsets. The term matroid arose from the generalisation

of the columns of a matrix following consideration of the independence

of those columns. This thesis is concerned with a class of matroids

M(E) which can be defined in a certain way from functions defined on

the ground set E .

It is well known that a matroid M(E) can be obtained from

submodular increasing functions defined on 2 E , but in practice such

functions are rather rare. The motivation for this thesis initially was

the hope that from the more prolific functions on domain E , it would

be possible in some way to build up to submodular functions which define

some well known matroids. This hope was partially fulfilled, but the

investigation uncovered a simply defined and interesting class, firmly

located in the usual hierarchies of matroids.

Matroid theory began in 1935 with Whitney's basic paper [27].

He had been working in graph theory for some years and had noticed

similarities between the ideas of independence and rank in graph:, theory,

and the ideas of linear independence and dimension in vector spaces,

and in this paper he used the concept of matroid to abstract and formalise

these similarities.

At about the same time van der Waerden [24] was approaching the

ideas of linear and algebraic dependence axiomatically, so he too was

instrumental in the birth of matroid theory.

After this bright beginning the study lapsed for about twenty years,

with the important exception of papers by Birkhoff 1 1, 1 MacLane D2]'P137,1 _—

and Dilworth [6 LE 7 ],[ 8] on lattice theoretic and geometric aspects

of matroid theory, and two papers by Rado [18],[19] on combinatorial

applications of matroids and infinite matroids. Tutte [21],[22]

vii

revived the study in 1958 with a characterisation of those matroids

which arise from graphs and at about the same time Rado [20] returned

to the field with a study of the representability problem of matroids

Since then interest in matroids and their applications has grown

rapidly and there is now a sizable body of literature on the subject.

Of particular importance has been the applications of matroids to

transversal theory and the associated investigation of transversal matroids.

This work was pioneered by Edmonds and Fulkerson [10] and Mirsky and

Perfect [17], and has produced many new results as well as elegant

proofs of earlier results in transversal theory.

Many other aspects of combinatorial theory have been subsumed in

matroid theory over the past 15 years and the result has been a

firmer linking of combinatorics to the mainstream of mathematics.

Matroids have been used for engineering applications recently, for

example Weinberg's work on electrical network synthesis [25].

The theme of this thesis, matroids defined by submodular functions,

had its beginning with a paper of Dilworth [ 8 ], in which seemed to be

implicit the fact that a matroid can be defined by its submodular rank

function. The first explicit derivation of matroids from submodular

functions is thought to be due to Edmonds and Rota [11] in 1966, and

a generalisation of this result was produced by McDiarmid [14]. Further

work on the relationship between submodular functions and matroids was

done by Edmonds [9 ] and Pym and Perfect [17].

In this thesis Chapter 1 is simply a restatement of •the many

different axiomatic ways of defining a matroid together with some well

known results necessary for the development of the thesis. A similar resume

can be found in a paper of Wilson [28].

Chapter 2 contains the basic "arithmetic" of the thesis. It

establishes that a submodular function on 2E

can be derived from any

integer valued increasing function defined on E and characterises the

matroids so formed by standardising the defining functions. The class

M of matroids so obtained isshown to be self dual and closed under

taking restrictions.

Whereas the treatment of the matroids of M was in arithmetical

terms in Chapter 2, in Chapter 3 the approach is more in the mainstream

of matroid theory. The class pl is characterised in terms of the

unique minimal non-trivial flats of its minors and also by its excluded

minors.

Chapter 4 shows that ri contains exactly 2 n pairwise non-

isomorphic members on an n-set. The number of matroids in the class

is compared to earlier lower bounds established by Crapo C 4] and

Bollobgs E 3 ] for the class of all matroids on an n-set. We then

examine those on a 6-set and by use of the excluded minor property the

matroids not in M are identified.

Chapter 5 establishes that M is properly contained in the class of

transversal matroids and obtains a necessary and sufficient condition

for a transversal matroid to belong to M . Another condition in terms.

of circuits is produced for a matroid to belong to M .

In Chapter 6 the results of the earlier chapters are extended to

the class of pregeometries, which are defined on possibly infinite

ground sets S by integer valued functions on S . The principal

tools in this investigation are the results on submodular functions (semi—

modular in E 5 A of Crapo and Rota, and Rado's Selection Principle.

The final chapter, Chapter 7, deals with functions defined on E r

from which submodular functions and ensuing matroids are obtained.

viii

ix

The value of the function on each r-tuple is the sum of the values of

functions (defined on E ) on the components of the r-tuple, and the

matroids so obtained are the union of matroids in M . As Welsh [26]

pointed out, this result is implicit in the more general results of

Pym and Perfect [17] for sums of arbitrary submodular functions. The

matroids so obtained are shown to constitute exactly

the class of transversal matroids. They are characterised in terms of

flats and circuits. We see that more graphic matroids are included in

this class of matroids than in M . Finally there is a failed

conjecture. It had been hoped that it would be possible to obtain

submodular functions defining some well known matroids from the functions

on Er

, by allowing r to increase. However a counter example is

provided.

With the exception of the abovementioned result implicit in work

by Pym and Perfect, the work in Chapters 2 to 7 inclusive is not

in the literature.

The author gratefully acknowledges the help of James Oxley and

Don Row in obtaining theorem 3.6, and also particularly James Oxley for

obtaining the excluded minor characterisation of theorem 3.9, and

suggesting a detailed examination of the members of M on 6-point

ground sets.

NOTATION

For most of the thesis we consider structures on a finite set,

and this set is designated E . When we deal with an infinite set

it is designated S . Elements of E or S are denoted by lower

case letters and subsets by upper case. The empty set is denoted by

(1) , and MB is the set consisting of elements which are in A but

not in B . AUB denotes the disjoint union of A and B.

Where the meaning is clear, we abbreviate {a} to a. For

example A u a means A u {a} and A\a means A\{a} .

A function from the set E to the set F is denoted by .

f:E F , and a function from the power set 2E

to the power set

2F

by 0:2E

2F .

If T is a subset of E we denote the restriction of f to T

by f IT . The matroid M on E restricted to T is denoted by MIT .

A family or collection of subsets of E is denoted by

(E.cE:E.Ims the required property), or, if it is possible 1 —

to list the subsets, by (E 1 ,E 2 , Em)•

The set of integers is denoted by Z .

CHAPTER 1

There are several equivalent ways in which matroids may be

defined. Proof of their equivalence is in Whitney's original paper

[27]. Some are listed below.

1. Independent Sets

This is the set of axioms favoured by many because of its obvious

relationship to linear algebra, which makes it easily recognized. It

is the set most commonly used in this thesis.

A matroid M(E, I) consists of a finite set E, together

with a non-empty collection I of subsets which are called

independent sets of E, satisfy the following properties:

(i) I E 1 and J c I .-->J E ;

(ii) if I,J E and IJI > III , then there exists a E J\I such that

IuaE I.

Any set not in I is dependent.

It follows from the above that all maximal independent subsets of

any subset A of E have the same number of elements. The

maximal independent sets are known as bases and their size is the rank

of the matroid. This brings us to the next two axiomatic descriptions.

2. Bases

• A matroid M(E, ID consists of a non-empty finite set E , together

with a non-empty collection B of subsets of •E , which are called

•

-- bas-es - ; satisfying the following property., ; .

if BB2

E $ and a E B1\B

2 there exists

such

that (B1 u b)\a E

1.

2.

3. Rank Function

A matroid M(E, p) consists of a non-empty finite set E , together

with an integer valued function p:2 E Z , called the rank function,

which satisfies the following properties:

(i) for each A c E , 0 p(A) IA1;

(ii) if AcBcE then p(A) p(B);

(iii) for any A,B c E , p(A) + p(B) p(A U B) p(A n B).

If p(A u a) = p(A), then a is said to depend on A , or to be

in the closure of A , and the set a(A) = {a E E: p(A u a) = p(A)1

is said to be the closure of A . This leads us to the next

axiomatic description.

4. Closure

A matroid M(E,a) consists of a non-empty finite set E , together

with a function (5:2E

2E

, called the closure operator, which satisfies

the following properties:

(i) for each AE. E,A=a(A) ;

(ii) if A a a(B) , then a(A) 5._a(B) ;

(iii) if a E a(A u b) , a a(A), then b E a(A u a) .

This is the set of axioms adopted by Crapo and Rota [ 5], and they

use the term pregeometry rather than matroid allowing E to be infinite.

The closures are also known as flats, and this term will sometimes be

used in this thesis.

The final set of axioms we consider is somewhat different, in that

it is not inspired by linear algebra but rather by graph theory. It is

in terms of circuits, which in graph theory are finite sequences of

distinct edges defined in terms of vertices as follows: {v 0 ,v 1 } ,

{v 1 ,v 2 } , {vm ,v 0 } , i.e. they are polygons. In a matroid a

circuit is defined as a minimal dependent set.

3.

5. Circuits

A matroid M(E, C) consists of a non-empty finite set E , together

with a collection C of non-empty subsets of E , called circuits,

satisfying the following properties:

(i) no circuit properly contains another circuit ;

(ii) if a E C 1 n C 2 , where C1,C2 E C

are distinct, then there

exists C E C such that C c (C1 2)\a'

This set of axioms was favoured by Tutte [23].

Throughout this thesis we do not distinguish between the sets

of axioms defining the matroid and merely represent it as M(E) .

As well as the matroid entities mentioned above, i.e. independent

sets, rank function, bases, closures and circuits, there are others

which are frequently used. Those which are used in this thesis are as

follows.

A cobase of the matroid M(E) is any set E\B, where B is a

base of M(E) . It can easily be shown that the collection of cobases

of M(E) is the collection of bases of a matroid, and this matroid is

designated M*(E) and is called the dual matroid of M(E). This

result was first established by Whitney [27].

Following from the above, the corank p*:2E Z of the matroid

M(E) is the rank function of M*(E).

A cocircuit of M(E) is a circuit of M*(E).

A hyperplane is a maximal proper flat of M(E) . It can be shown

that a hyperplane is the set complement of a cocircuit.

We now consider a few types of matroids which will be referred to

later. They are graphic matroids, transversal matroids and matroids

representable in Euclidean space. First we need a definition of isomorphism

of matroids.

4.

Two matroids M1(E

1) and M

2(E

2) are isomorphic if there exists

a bijection 0:E 1 -0- E 2 which preserves independence.

A graphic matroid is one which is isomorphic to a matroid defined on

edges of a graph by letting the circuits of the matroid be the edge

sets of polygons of the graph.

A transversal of a finite family U = (E 1 ,E 2 ,..., Em ) of subsets

of E is a set of m distinct elements of E , one chosen from each

ofthesubsetsE.;a partial transversa of U is a transversal of

some subfamily of U • It is easily shown that the partial transversals

of U satisfy the properties specified above for independent sets of

a matroid. The bases of the matroid are the maximal partial transversals

of U . We call a matroid M(E) a transversal matroid if there exists

some family U of subsets of E such that the family of independent sets

of M(E) is precisely the family of partial transversals of U

Euclidean representation of a matroid is possible if it is

isomorphic to the matroid induced on a set of points in R n by

the usual affine closure.

As we saw above, a function p:2 E Z having certain properties

defines a matroid whose rank function is p . One of those properties

was that for any A,B c E , p(A) + p(B) p(A u B) + p(A n B) , and

• a function having this property is known as a submodular function.

A function having the property that AcBcE-1>p(A) p(B) is an

increasing function. The following observation is used throughout this

thesis.

THEOREM 1.1 A submodular increasing function f:2E Z defines a

matroid on the set E .

5.

Proof: Let C = (q) C E 2E : f(C) < ICI, f(K) (KI for all K c C).

We proceed to prove that C is the collection of circuits of a matroid.

(i) Obviously no member of C properly contains another member

of C.

(ii) For any C E C s f(C) = ICI -1. We consider now any distinct

CC2

E C whose intersection is non-empty, containing say the

element a. Now applying the submodularity of f, and the fact that it

is increasing, we have

f((C i u C 2 )\a) f(C i u C 2 ) f(C 1 ) + f(C 2 ) - f(C i n C 2 )

1C 1 1 7 1 + 1C 2 1-1 - 1C 1 n C 2 1

< 1(C1 u C 2 )\al .

Furthermore we know that (C1

u C2)\a contains a set K such that

f(J) IJI for all J c K , since at least ya and C 2\a have

this property. Therefore (C 1 U C 2 )\a contains a member of the

collection C , whence C is the collection of circuits of a matroid. //

COROLLARY 1.2 The collection 1 of independent sets of a matroid

M(E) defined by a submodular increasing function fp2 E -* Z , is

given by

= {I E E: f(J) IJI for all J c I} u . //

The above corollary appeared in a paper by Pym and Perfect [17]

in 1970. As remarked in that paper, Edmonds and Rota had already

proved a more comprehensive result.

It is necessary to point out, as did Pym and Perfect [17], that if

M(E) is a matroid on E , it may be possible to find a function

f:2E

Z which is not submodular, but for which the set I is

independent if and only if f(J) IJI for all J c I . Their example

was as follows. Let M(E) be the free matroid on E , and define

f:2E Z by the equations (i) f(A) = 1E1 for A c E, and

(ii) f(E) = 21E 1. We recall from graph theory that a graph can have a loop, i.e. an

edge whose two vertices are identical, and multiple edges, i.e. edges having

the same two vertices. A graph having no loops or_multiple edges is

called a simple graph. Analogously a matroid can have elements of rank

zero, i.e. they are in the closure of the empty set, and it can have a

set A such that At > 1 , p(a) = 1 for all a ,E A , and p(A) = 1.

A matroid having neither of the above is a simple matroid. Obviously

a graphic matroid is simple if and only if it is isomorphic to a matroid

defined in the abovementioned manner on a simple graph.

6.

CHAPTER 2

In the previous chapter we saw that an integer-valued increasing

submodular function on a set E defines a matroid. Examples of

such functions are the dimension function on subspaces of a vector

space (in which the submodular inequality becomes an equality and

the function is modular), and the rank function of a matroid. In

this chapter we construct a submodular function on arbitrary sized

subsets from a function defined on singletons, and in this manner-

generate a particular class of matroids.

We obtain the function f : 2E

Z from a function p : E Z

as follows. Let

f(A) = max{p(a):a E A} for all (I) A E 2E ,

and f(A) = min{p(a):a E E} for A =

LEMMA 2.1 Let f be a function as defined above. Then f is

increasing and submodular.

Proof: It is obvious that f is increasing. For any subsets

A,B c E let a E A, b E B be such that p(a) _ p(x) for all x € A,

p(b) p(x) for all x € B. If p(a) p(b) then f(A) = f(A U B)

and f(B) f(A n B), whence f(A) + f(B) f(A U B) f(A n B).

The same result holds if p(b) p(a).. //

We call matroids induced by the functionlas defined above

matroida of the class M , and similarly f is called a function

of the class F .

The 9eometric structure of the matroid i characterised by its

independent set, closures, circuits, bases, and we now examine

7.

some of these for the matroids of class M .

The independent sets of a matroid obtained from a submodular

function f are the family I of sets given by

= (I : f(J) IJI for all J = I) .

In terms of p , 1 is given by the following lemma.

LEMMA 2.2 If M(E) E M is obtained from the function p : E Z,

then the collection of independent sets of M(E) is precisely the

family

l'Ou(I=E:otJ=I*3acJsuchthatp(a)IJI).

Proof: Suppose I is independent, i.e. 'f(J) IJI for all J = I.

Then on any J = I the maximum value p takes is at least IJI ,

so there exists a E J such that p(a) IJI.

Conversely suppose ; then obviously.for every J = I,

f(J) , whence I is independent. //

A description of the circuits of a matroid of the class is given

in the following lemma.

LEMMA 2.3 If M(E) E M is obtained from the function p : E Z,

a subset C = E is a circuit of M(E) if and only if its elements

can be labelled a l, a r (where ICI = r) such that

r - 1 u(a) i for all 1 i r-1, and p(ar) = r-1.

Proof: Suppose the elements of C can be so labelled. Then

obviously f(D) IDI for all D = C\Ia r l . Further f(D) IDI

for all Dc C\{a.} where i r, since this set is obtained from —

8.

9.

C - ia rl by substituting a r for a i , and p(a r ) p(a) .

Therefore f(D) 'DI for all D c C and f(C) < ICI so C is a

circuit.

Conversely suppose C is a circuit; then p(c) < r = ICI for

all c E C. Let ar

be an element of C on which p takes its

maximum value, which necessarily is r - 1. Let a r-1 be an element

of C\{a r } on which p takes its maximum value, which is r - 1.

By continuing this process we obtain a r-2' al so that

p(a 1 ) i for r - 1 i 1.

The closure a(A) of a subset A c E is given in the following

lemma.

LEMMA 2.4 The closure G(A) of a subset A c E in the matroid

M(E) E M is given by

a(A) = A u {a E : p(a) IJI} ,

where J is a subset of A maximum with respect to

(i) J is independent, and

(ii) mxiii -(a) . : a = [J].

Proof: We show that a(A) as defined above is precisely the cldsure

of A in the matroid M(E). Obviously any b E A is in both the closure

and a(A). Consider b / A but in the closure of A. Then the joining

of b to A does not increase the size of any maximal independent set

in A, whence p(b) IJI and b E a(A). Conversely suppose

b / A, b E G(A). Then p(b) 5_ IJI, whence b does not increase the size

of any maximal independent set in A, and b is in the closure of A.

'Therefore a(A) and the closure of A in M(E) are identical. //

Information about the matroids under consideration is more

accessible if some ordering exists on the elements of E. Since this

only involves relabelling the elements, no generality is lost. We say

that E is p-ordered when the elements of E are arranged and

identified by the symbols a l , ..., a n (where 1E1 = n) such that

p(a i+1 ) p(a) for i = 1, ..., n-1.

Similarly a set A c E is p-ordered if E is p-ordered, and we

identify the elements of A as a11'

..., aim a where il < ... < im.

We now move on to a consideration of bases and we recall that a

basis is a maximal independent set.

LEMMA 2.5 Any maximal set {aij : j, j = 1, ..., r}

of the p-ordered set E is a basis of the matroid M(E) E M.

Proof: The set is obviously independent and also maximal. //

LEMMA 2.6 If there exist a n , a n _ i , a n _ s E E such that

p(a ) > > p(a n _ s ) > p(b) for all b E E\{a n , a n _ s }

then a n , ..., a n _ s are in every basis.

Proof: Suppose the result is true for a n , ..., a n _ j , where j < s.

We proceed by induction on j. Suppose an-j-1 is not a member of

every basis and let B be a basis such that B. Then

a n _j 1) u {a n _ j _ 1 }) > a n _j } _ 1BI - j - 1,

10.

11.

whence f(B U lan-j-1

1) IBI + 1, which contradicts the maximality

of B with respect to independence. Therefore if the result is true

for n, n-j , it is also true for n - j - 1.

We now consider the case of j = 0. Suppose a n B, where B

is some basis. Then f(B U an) > f(B) IBI , whence B U a

n is

independent, which contradicts the maximality of B. //

It is obvious that there are many functions p which induce

the same matroid, so we now find upper and lower bounds for all such

functions and establish their uniqueness. We begin with a standardised

function obtained from p . This standardised function 1 is defined

as follows on a p-ordered set E:

1(a k ) = min(p(a k ), 1(a k _ i ) + 1), where

1(a 1 ) = max(min(1, p(a 1 )), 0) .

We sometimes say that a k is a member of level 1 or has

level 1(a k ).

LEMMA 2.7 If l(a) = 1(a k ) for elements a j , a k of a. p-ordered

set E and j < k, then p(a j )=p(a k ) = l(a) = 1(a k ).

Proof: Since j < k we have l(a)

l(a) whence

l(a) < 1(a k _ i ) + 1. Therefore l(a) = 1(a k ) implies that

1(a k ) = p(a k ) , and since' 1(a j ) p(a j ) p(a) we have the result. //

LEMMA 2.8 The rank p(M) of the matroid M(E) E M on a p-ordered

set E is given by p(M) = max{1(a): a E E} .

Proof: If max{1(a): a E E} = r then 1 takes all values from 1 to r

(and possibly also 0), and only those values. Hence we can choose

12.

a set B = {13 1 , br } c B so that 1(b i ) = i for all

1 i r. Then p(b i ) i and hence B is independent. The function

1 maps any other a E E to say k, and this property is shared by

bk

E B. By Lemma 2.7 therefore p(a) = p(bk) whence the set

b k , a1 is not independent. B is therefore a basis. //

LEMMA 2.9 A set comprised of single representatives of any number

of distinct levels is independent. //

LEMMA 2.10 A p-ordered set B = {a11 ,

• .., a. } is a basis of a ip

rankpmatroidricrlifandorayil-1( a l ) 1.=p and 1(a..) j

pJ

for all 1 j p- 1.

Proof:Supposel(a.i )=pand j for 1 5_ j p-i; then

p 1J

obviously {aij: 1 5_ j 0 is independent and being of size p must

be a basis. Conversely suppose B is a basis; then 1(a il ) 1 and

also 1(aij

) ..Zj for all 2 -.:sp , otherwise there exists j such

thatl(a.1.)=j-1andl(a ij )= 1(a 1 ). The latter implies J

that p(a) = j - 1 which contradicts the independence of B.

Lemma 2.8 establishes that 1(a 1p )= p . //

The function 1 is defined on all elements of E, and E is 1-ordered

n the same sense as it is p-ordered), so it is natural to enquire

what standardised function is obtained from 1. It turns out that

the process of standardising the function is an idempotent process

as can be seen from the following definition and lemma. We define

12(a

k) = min(1(a ), 1

2(a k _ 1 )+1) and

12(a 1 ) = max(min(1, 1(a 1 )), 0) ,

2 is the standardised standardised function obtained from p.

13.

LEMMA 2.11 12

= 1 .

Proof: We proceed by induction on k. Suppose 1 2(a k ) = 1(a k ) ; then

12(a k+1 ) = min(1( k 1

) ' 12(a k )+1)

= min(1(a k+1 ), 1(a k )+1)

= Maw. ) .

It is obvious that 12(a 1 ) = 1(a 1 ). //

LEMMA 2.12 induces the same matroid M(E) E M as does p .

Proof: 1 induces a matroid of the class by Lemma 2.1 and according

to Lemmas 2.10 and 2.11 the matroid has the same bases as that

induced by p . //

LEMMA 2.13 If 1,h: E Z are standardised functions obtained from

p,v: E Z respectively, and p,v induce the same matroid

M(E) E M ,then 1 = h.

Proof:Let.Ebep-orderedandleta.1 be the first element of E

forwhichl(a.)# h(a.1). Suppose 1(a.) > h(a

1.) and 1(a.) = k.

1 1 1.

Then single representatives from each of the 1-levels 1, ...,,k-1, with ai'

constitute an independent set in the matroid induced by p , but

a dependent set in the matroid induced by v . A similar result follows

ifh(a.)>1(a.). Since both matroids are the same we conclude

1 = h. //

In summary, all functions p: E Z which induce the same matroid

• M E M effect the same standardised function1: E Z, and 1

induces the matroid M also.

14.

We move now to another function obtained from p. It is apparent

that any value of p(a) in excess of p(M) has no effect on the

structure of the matroid. Therefore we normalise p in the following

way:

= P(a), if 11 (a) < P(M) 0(a)

= P(M), if P(a) P(M)

We also define a further function p': E Z, which is obtained

from p , as follows:

= max fp(b): b E C1, where C is a circuit

of maximum cardinality containing a , p 1 (a)

= p(M) , if a is not in any circuit.

.LEMMA 2.14 The matroid induced by p' is precisely the matroid

induced by p.

Proof: Let I . be independent in M(E), the matroid induced by p.

Suppose there exists a E I such that a is not a member of any

circuit; then

v(I) = MaXi11 1 00:X E 11 = P(M) III .

_ For those subsets J c I whose elements are all members of some

circuit , f(J) IJI implies that some b E J is in a circuit of

cardinality at least IJI + 1, whence p 1 (b) IJI and -V(J) IJI.

Therefore fi(J) IJI for all J c I and I is independent in

the matrOd induced by p'.

Conversely let I be independent in the matroid induced by p'

and suppose that for some J c I, f(J) < IJI. If J is not minimal

with respect to this property we choose J l which is. Then J l is a

circuit of M(E), which implies that for all b E J 1 , P I M < 011,

15.

whence I is not independent in the matroid induced by p' and we

have a contradiction. Therefore there cannot be any J c I for which

f(J) < ljl and so I is independent in M(E). //

LEMMA 2.15 If p, v induce the same matroid M E M, then p' = v'. //

It has therefore been established that for a matroid of class m, both 1 and p' are unique. The following lemma shows that they

are lower and upper bounds of all the normalised functions which induce

the same matroid, i.e. they are unique lower and upper bounds.

LEMMA 2.16. 1 p.

Proof: The first part is obvious from the definitions_ of 1 and 1.

For the second, j(a) = p'(a) for all a such that p(a) - p(M) and

p(a) = p(a) p 1 (a) for all a such that p(a) < p(M). //

The uniqueness of p' may be expressed in terms of the auto-

morphisms of the matroid. We define an automorphism of the matroid

; M to be a bijection 0:E+ E such that I is independent in M

7 f and only if 01 is independent in M. (Here 01 means {0(a): a E 1 }-)i

I;

LEMMA 2.17 The automorphisms of a matroid M(E) E m . are precisely the p'-preserving permutations of E.

Proof: This follows immediately from the uniqueness of p'. //

We return now to a study of the standardised functions 1.

A graph of 1 against the elements of the p-ordered set E is

revealing because it pictorially] conveys information about the

structure of the induced matroid. Bases, circuits and closures are

more easily discerned. An example of a graphical matroid which

belongs to the class M is depicted below.

16.

.9

Fig. 1. Graph of function 1 for a graphical matroid.

-

Another way in which matroids may be characterised is by means

of cocircuits, and we use this characterisation to move towards duals

of matroids of class M. A cocircuit is the set complement of a

hyperplane and therefore the cocircuit can be described in terms of a

basis and a single element of that basis. If B i is a basis and

a.. B. then we denote the associated cocircuit as D. .. This description 1

need not be unique, but every cocircuit can be so described.

LEMMA 2.18 If B i = {a il , aip } is a p-ordered basis of the

matroid M(E) E M , and a ii E B. then the cocircuit D ij is given

by

= (fa: 1(a) > m(k < j: p(a ik ) = k)}\B i ) U {a ij }

where

k < j: p(a ik ) = k) = max(k < j: p(a ik ) = k) if k exists,

= 0 if no such k exists.

Proof:ThehyperplaneobtainedfromB.and aij

falls into one of

two classes, namely (i) those for which there exists k < j such

that p(a ik ) = k , and (ii) those for which p(a ik ) > k for all

1 5 k 5 j. For (i), Lemmas 2.4 and 2.10 establish that

= (B.\{a..}) u {a: 1(a) 5 max(k < j: p(aik

) = k} , 1 1J 1 1J

and the complementary cocircuit is as required.

For (ii), a(B 1 \{a..}) = B.\{a..} and therefore 1J 1 1J

= (E\B.) u {a.1.} , which can be rewritten

1 J

D i j = 1(a) " 1 1J

We pow introduce another function derived from 1 , which will

be necessary in obtaining the dual matroid. We define 1*: E 4- Z,

where 1E1 = n and p is the rank of the matroid induced by 1 , as

follows:

1*(a i ) = 1(a i _ 1 ) +n-i+ 1 - p

and 1*(a 1 ) = n - p

The following res6lts are necessary in establishing duality.

LEMMA 2.19 (i) 1*(a 1 ) = 1*(a.0.1 ) if 1(a i ) = 1(a i _ 1 ) + 1

and 1*(a 1 ) = 1*(a.0.1 ) + 1 if l(a) = 1(a i _ i )

> j * 1*(a 1 ) 5 1*(a j )

and i > j 1*(a i ) < 1*(a j ) . //

It is obvious that 1* induces a matroid of class M on E,

and we denote this matroid by M l . By the reasoning of Lemma 2.8,

P(M 1 ) = 1*(a1

) = n - p .

LEMMA 2.20 1** = 1.

Proof: We 1*-order the elements of E by reversing the p-order.

This is consistent with Lemma 2.19. Then for any a i E E

(i being the position in the p-ordering), we have

17.

II

1** = 1*(a ii.1 ) + n - (n-i+1) + 1 - (n-p)

= 1*(a. ) + i - n + p, 1+1

and by substituting for 1*(a1+1

) we complete the proof.

LEMMA 2.21 If Ml (E) is the matroid induced by I* , then the

levels of the elements of E in M1(E) are the values of I*

on the elements.

Proof: Let the standardised function obtained from 1* be L,

4nd let the 1*-Ordering be the reverse of the p-ordering. Then for P

all a i E E, L(a) = min(1*(a i ), 1*(a i+1 ) + 1) = 1*(a 1 ) by Lemma 2.19. //

_

We come now to the most important result of this-chapter, namely ,

that the class M is closed under taking duals. We use the fact

that one matroid is the dual of the other if and only if the circuits

of one are precisely the cocircuits of the other.

THEOREM 2.22 M* = M .

Proof: We can assume, without loss of generality, that E is

it-ordered. We refer throughout to levels in Ni and M 1 and to

avoic confusion we call them 1-levels and 1*-levels respectively.

Lemma 2.19 implies that the 1*-levels have the following

structure. Elements on a particular 1-level in M occupy, in reverse

u-order, successive 1*-levels, except for the first element in that

1-level, which occupies the same 1*-level as the second element in the •

18.

1 /

19.

non-z ero

1-level. All elements in successive single element 1-levels occupy A

the same 1*-level, and this 1*-level is that of the first element of

the multi element 1-level immediately greater than them. If there is

no multi element 1-level'greater than the abovementioned single

element 1-levels, then 1(an) = p and l(a 1 ) = p - 1 , whence

1*(an) = 0 and the elements of all those single element 1-levels

occupy 1*-level 0.

Consider a cocircuit D ij determined by the basis

13.--{a1' "" . a.}andtheelementa j . .Let

1

m(k < j: p(a ik ) = k) = h and let m be such that p(a m ) = h

and p(amil ) > h. Then

= (ia u } , m+1' a n l\fa

i(h+1)' aip/) u {a

and it hasn+h+ 1 -m-r elements. Further, the maximum value

of 1* on {am+1 , ..., a n } is 1*(am+1 ) =h+n-m-r, and

also by the reasoning above on the 1*-levels, 1*(a..1J ) =h+n-m- r.

All we now require for D ij to be a circuit of M 1 , is for the

value of 1* on the n+h-m-r elements of D..\a.. arranged 1J 1J

in reverse p-order to be at least 1, 2, ..., n+h-m-r respectively.

It is obvious that representatives of each of the l— levels

1, 2, ..., n+h-m-r have the required property. If therefore

ak (k j) is the lone, first or second element of the k-th

i

1-level its removal from lam'

..., an

still leaves a representative

of its 1*-level. If 1(a ik ) = k but a ik is not the lone, first

or second element of the k-th 1-level then its removal from

..., a n } also removes an 1*-level, but this is compensated

for by the first or second element of the k-th 1-level, on which the

value. of 1* is higher than on a ik . Finally for each a ik such

20.

that 1(aik

) > k , there exists a lesser 1-level (but greater than

h) which does not have a representative in B i , and therefore on

the lone, first or second element of that 1-level the value of 1*

is at least 1*(aik'

) and therefore compensates for the removal of

a.k . i

. This establishes that the removal of a.k

k j , h+1 k p i'

from {am+1 , ..., a n } leaves n+h+l-m-r elements which constitute

a circuit in MI(E).

We now show that every circuit in M1(E) is a cocircuit in

M(E). Let L4 = fa E E: 1(a) = i} for all I i p and 'HI = n i .

Further, let C* be a circuit in M i and let L p be the 1-level

containing the first member of C* , i.e. the element having the

lowest subscript. Since the value of 1* on the first and second

elements of C* is the same, it follows that the first element is the

first element in the 1-level p or is a single element 1-level.

TherankofC*inM l isthen n i l.-(p-p+1) and the number i=p

of.elements in C* is X.n 4 - (p-p). i=p

Let the h-th 1-level be the greatest multi element 1-level less

than p (we take h = 0 if no such level exists). Then the number

of elements of 1-level greater than h is n. and the number of i=h+1 1

P those not in C* is 1 n. - n. + p - p , which equals p - h - 1.

i=h+1 • i=p 1

C* is a cocircuit if the above p - h - 1 elements, together with

an element of C* , belong to a basis of M. This element of C*

must not have a multi element 1-level between its 1-level and 1-level

h.

Let the g-th 1-level be the lowest 1-level greater than h which

has more than one element. If no such 1-level exists then 1*(a) = 0

for all a such that 1(a) > h and all such a are single element

21.

circuits of M 1 . It is obvious that they are single element cocircuits

of M , and for this case the proof is complete.

If such an 1-level does exist we consider the 1-levels

q+1, p . Suppose that for any j , q+1 5 j 5 p , less than

p -j + 1 elements of .'L U U L are excluded from C* , i.e.

atleast ilMnbersofClrarein" u Lp

.

Since the maximum value of 1* on this union is n ; - (p-j+1),

this implies that C* properly contains another circuit of M* ,

which is impossible, so we conclude that at least p - j + 1 elements

of L U L are excluded from C* for ci + 1 5 j 5 p .

We now have that the number of elements of C* is

n + (nq+1-1) + + (n -1). Further, 1-levels q to. p inclusive

contain an independent set in M , disjoint with C* , of size at

least p-q , and there are q-h4-1 single element 1-levels between

1-level h and 1-level q. There are three possibilities for the

composition of C* , namely:

( ) C* contains all of 1-level q and none of the elements from

the single element 1-levels between h and q.

C* contains all of 1-level q and some of the elements

from the single element 1-levels between h and q.

(iii) C* does not contain all of the elements from 1-level q,

which implies that it must contain some elements from the

single element 1-levels between h and q.

If possibility (i) applies then the p-q elements of 1-levels

q+1 to p inclusive which were omitted from C* , together with

the q-h-1 elements between 1-levels h and q, and any element

from 1-level 1, form part of a basis, and hence C* is a cocircuit of M.

22.

If possibility (ii) applies then p-q elements of 1-levels

q+1 to p inclusive are not in C* and for every element of 1-levels

between h and el which is in C* there is an additional element

from 1-levels q+1 to r inclusive not in C*. These elements not

in C*, together with the elements between 1-levels h and q not

in C* , and any element from 1-level q , form part of a basis and hence

C* is a cocircuit of M.

Finally if possibility (iii) applied then p-q+1 elements of

1-levels q to p inclusive are not in C* , and for each element

of 1-levels between h and q which is in C* there is an element

from 1-levels q to p inclusive not in C* . These elements not

in C* , and one element between 1-levels h and q which is in C* ,

form part of a basis and hence C* is a cocircuit of M. //

It is informative to look at the graphs of 1 and 1* and the

figure below is an example.

/0

1.5"

/0

Fig. 2. Graphs of 1 and 1* on a 15 element set.

23.

It will be noted that there is a relationship between the

gradients of the two graphs. For example the gradient of the

function 1 for 1 5 i 5 4 is 0, whereas that for 1* for

2 5 i 5 5 is -1, and the gradient of 1 for 8 5 i 5 12 is 1

whereas that for 1* for 95i 5 13 is 0. Inspection of the

relationship between 1 and 1* shows that this is general, i.e.

l'(a .) = 1 for i j k 1* 1 (a.) = 0 for i+1 j k+1 j

and1 1 (a.)=. 0fori "(a j.)=-1 for i+1 < j 5 k+1.

Another way of viewing the above is to represent the set as

in the figure below. A 1*

9- 21

8 - 3

A1 7- 4

12 13 14 15 6- 65

11 7

4— 10 4- 8

3— g 3 - 13 12 11 10 9

5678 2 -

2- 14

/- 1 2 3 4 /- 15

1-level s 1*-levels

Fig. 3. Levels of 1 and

In the above figure rows can be regarded as comprising elements

for which there is no increase in level over the preceding element,

while columns comprise those for which there is an increase in level

over the preceding element. With this classification rows in 1

representation are columns in 1* representation and vice versa.

Again, because of the relationship between 1 and 1* , this

result is general.

We conclude this chapter with a lemma concerning restriction.

24.

LEMMA 2.23. M IT = M , where the subscripts refer to the P IT

inducing functions of the matroid of class M

Proof: We define f iT (A) = max{u lT(a): a E A} for all A c T .

It is immediately obvious thaton all AcT, f IT = f. It follows

then that for any I c T which is independent in M , f iT (j) IJI

for all J = I , whence I independent in MIT I independent in

P IT

Conversely if I, is independent in M then f(J) = fIT

(J) IJI IT

for all J = 1 and also I = T , whence I is independent in

M IT . / /

25.

CHAPTER 3

Chapter 2 was concerned with the %rithmetic" of matroids of fol .

We now, establish that a characterisation in more general terms is

available. Firstly we show that M consists exactly of matroids, all of

whose minors are free or have unique minimal non-trivial flats.

Secondly we give an excluded minor characterisation of M. Again in

this chapter E is finite. The term flat rather than closure in used

so that we can conveniently speak of it without reference to the sets

of which it is the closure. A flat F of M is non-trivial if it is

the closure of a proper subset. It is a non-trivial extension of a flat

H if it is the closure of H u P for some proper subset, P of

F\H. Otherwise F is a free extension of H.

Consider a matroid each of whose minors is either free or has a

unique minimal non-trivial flat. We denote the class of matroids

having this property by M'.

LEMMA 3.1 Each M I EM I on a ground set E, has a 'finite chain

a(cp) = Fo c F 1 ... c F k c E, where F.0.1 is the unique minimal

non-trivial extension of F i for 0 i < k and F k has no non-trivial

extension. Each flat in M' is a direct sum of some F. and a

free matroid.

Proof. Let cy((p) = F o and suppose there exists a chain F o c F 1 c F.

• such that Fj+1

is the unique minimal non-trivial extension of F.

for

0 j < i. Then either E is a free extension of F. in which case

k = i, or there exists a minimal non-trivial extension of F. . Suppose

there exist two such extensions H and H' . Then we consider the

minor M' o (E\(H n H')).

26.

Applying a standard result of matroid theory we have x c acont

(H\H')

x E a((H\H') U (H n H')) = H. Since the minor is a matroid only

on E\(H n H') we conclude x C acont("1) and x

E E\(H n H')

4P X E H\H' , whence H\H' is a flat in the contraction.

AsHisanon-trivialextensionofF.in M' it contains a

circuitCwhichisnotcontainedinF..Furthermore H n H'

either is F. or is a free extension of it, whence C ¢ H n H',

so H\H' contains a circuit in the contraction. Therefore H\H'

is a non-trivial flat in the contraction, and by the same reasoning

so is H'\H. It follows that both contain minimal non-trivial flats

which must be disjoint. This is impossible since M' E M' and we

conclude that there exists F1+1

which is a unique minimal non-trivial

extensionofF.. By induction we obtain the required chain of flats.

Any flat either (i) is free, or (ii) is an F i , or (iii) is a

free extension of an F. . Therefore a flat F is the direct sum

.ofF.,forsome05.k,andthefreematroid //

We prove M' c M by characterising the circuits of members of

LEMMA 3.2 For any M I E M I , having flats as specified in Lema3.1,thecircuitscontainedin.but not in F i F are

exactly C satisfying ICI = p(F i ) + 1, IC n F i l p(F) for j

These, for all i, are the circuits of NV.

Proof: We proceed by induction. Either F o = (p. whence the circuits in

F 1 have the required properties, or each element of F o is a loop C

satisfying ICI = 1 = p(F0 ) + 1. Now suppose the circuits contained in F j

but not inF j _ l are as prescribed for all j < i.

27.

If C is a circuit contained in F i , C F i _ l , then o(C) is a flat

which by Lemma 3.1 isF. some j and obviously j = i ; therefore J'

independent

and IC 11 F.1 = g" P(F

Conversely let C satisfy C c F i , C ¢ F i _ 1 , ICI = p(F i ) + 1,

IC nF j 1 p(F) for all j < i. From this prescription C is

dependent and so contains a circuit C' . If C' c F. for some — J

j < I , IC' 1 = IC' n Fi l IC n F i l p(F) , which implies that

IC' l p(F) + 1, contradicting the proven property of any such

circuit. Hence C' c F i , C' ¢ F 1 _ 1 , so IC'I = p(F 1 ) + 1 = ICI, and C = C' .

We have inductively characterised all circuits contained in some

F i . But every flat is the direct sum of some F i and a free matroid,

hence all circuits have been characterised. //

LEMMA 3.3 M' _c_M •

Proof: Consider any M' E M i with a chain of non-trivial extensions

as specified in Lemma 3.1. We define an appropriate function on the

ground set E of M' as follows:

if e E F.\F1' with F ..1 = (I) i-

p(e) = p(E), if e E\Fk .

The function p induces a matroid M E M and we prove M = M'

by considering the circuits in both.

If C is a circuit in M' then for some i, C c F. ,

C ¢ F 1 _ 1 , ICI = p(F i )+1 and IC n Fl p(F) for all j < i.

Let C = ic l , c s } where p(c 1 ) . . p(c 5 ) = P(F) = ICI - 1 = S 1'

28.

For all r < s, either Cr e F1\Fi-1' or Cr E Fj\Fj-1

for some j < i.

In the first case, u(c r ) = P(F) = s - 1, and in the second case

p(c r ) = ()(F.) ?_ IC n F I r. We conclude that s - 1 p(cr

) r

for 1 i s-1 , and p(cs = s - 1 , and so by Lemma 2.3 C is a

circuit in M .

I Conversely if C is a circuit in M p , S-1 ?.. p(c) minfr,s-11

for 1 5. r s = ICI and so p(c 5 ) = s-1 = p(F i ), say: Then for

F .

, giving r J

iCnFH10.(F..).--But-s 7 1C1=10(FJ 1- 1: Hence C is a circuit j J 1

In M' . //• L._

To prove M = M' it suffices ifwe prove that M E M is

- either free or has a unique minimal non-trivial flat, and that M

is closed with respect to taking minors.

LEMMA 3.4 Each M E ri is a free matroid or has a unique minimal

non- trivial flat.

Proof: Let F and F' be minimal non-trivial flats in M with

p(F) p(F 1 ). Lemma 2.4 implies that p(a) p(F) for all a E F

and p(a) p(F 1 ) for all a E F'. It follows also from Lemma 2.4

that F = F' and since both are minimal, F = F'. //

We know from Lemma 2.23 that M is closed with respect to

restrictions and it remains to show that the same applies for

contractions.

LEMMA 3.5 Any contraction of a member of N is in N.

Proof: If M(E)EM, then for any T=E ,M.T= (WIT)*

and we know from Theorem 2.22 that M* E M .

29.

THEOREM 3.6 M = M' 0 //

We move on now to the second part of this chapter, namely the

excluded minor characterisation. We characterise mi and hence Ni

by its excluded minors. For k = 2, 3, ... consider a set E,

1E1 = 2k , E = E l i E2 with 1E 1 1 = 1E 2 1 =k and put -

C = {E l , E 2 } u tC: C E l , C E 2 , C c E, 1C1 = k+1}

. LEMMA 3.7 C is the collection of circuits o

k a matroid M wi.th

underlying set . E, for each k = 2, 3, ... .

Proof: Consider any two distinct members C1'

C2 of C with a

common element e. Then 1(C 1 u C2 )\el k+1 and so (C 1 u C 2 )\e

contains a member of C

//

LEMMA 3.8 k

Mi M' .

Proof: Both

and E2

are minimal non-trivial flats. //

THEOREM 3.9 M' is characterised by the family kM k = • • •

of excluded minors.

Proof: We consider any matroid which is not in fir ; it has at

least one minor which has two minimal non-trivial flats. We choose proper

a minor M which satisfies this condition but whose own minors are A

in M' . This is possible since, if not, the matroid has no minor

which has a unique minimal non-trivial flat or is free, and minors

of rank 1 obviously have this property.

The chosen minpr has two minimal non-trivial flats, say E l nAtIAA•mx) Nrcok

and E2 ,\ . If . E E l u E2 we choose e E EqE1 U E 2 ) and obtain

the restriction M1E\e. Since I independent in MI independent

in M1E\e for I c E i , i = 1 or 2 , it folloWs that E l and E 2

are minimal non-trivial flits in M1E\e. But this is

30.

a contradiction of our choice of minor. Thus E = E

We now prove that E l and E 2 are circuits of M. Suppose

E1

is not a circuit; then M has a circuit C which is properly

contained in E l and we consider the contraction .M.(E\e) where

e E E1\C . This contraction has non-trivial flats E

1\e

and E

2

(or E2\e if e E E

2) and these are minimal, which is a contradiction.

Therefore E l is a circuit, and similarly E 2 is a circuit.

This paragraph shows that E l and E 2 are disjoint. We assume

to the contrary that e E E l n E 2 , and consider the contraction

Mo(E\e) . In this ye and E 2\e are both circuits and flats,

and hence minimal non-trivial flats. Therefore ye = E 2\e ,

whence E1

= E2 '

which contradicts our choice of M. Hence

and E 2 are disjoint.

We show that 1E 1 1 = 1E 2 1. Choose any element in E , say

e E E2 '

and form the minor Mo(E\e). In this contraction E...\e c

is a circuit and a flat, and hence a minimal non-trivial flat.

Also acont

(E1) = 'a(E

1 U e)\e is a non-trivial flat, and by the

choice of M , cannot be minimal. Therefore

pcont

(E2\e)

< ' o whence p(E 2 )-1 < p(E 1 u e)-1 = p(E 1 ) , - cont (E 1 )

since e E l = a(E 1 ) , and it follows that p(E 2 ) p(E 1 ) . Choice

of any a E E l similarly leads to p(E 1 ) 5 p(E 2 ) , and we conclude

that 1E 1 1 = 1E 2 1 = k , say, for some k > 1.

It only remains to prove that the circuits other than E l and

E2

in M are exactly the subsets of E of size k+1 which contain

neither E1

nor E2 •

Since E1

and E 2 are minimal non-trivial

flats of MI it follows that all circuits have at least: k elements.

Suppose C is a third circuit of M and ICI = k; then C n E l 0

31.

and C n E2

0 . The flat q(C) is non-trivial with rank k-1, and

it has a subset F which is a minimal non-trivial flat. Considering

the minimal non-trivial flats E1

and F as above, we have

E = El 0 F , F is a circuit and IFI = k . Therefore F = C and

C n El = 0 which contradicts the necessary properties of C, and so

k+1 . We need only to show that p(M) = k to prove that all

circuits other than E l and E 2 have size k+1. Choosing e E E 2

and considering the contraction Mo(E\e) as above, we have

E 2 \e c acont(E1) = a(E 1 u e)\e , whence E 2 c a(E 1 u e) and so

El u e spans M , giving p(M) = k. Consequently M =

kM , for

some k > 1. //

32. CHAPTER 4

The numbers of simple matroids on ground sets of small sizes are

well known 2], and using this information it is easy to find the

numbers of matroids on those sets. It is natural to enquire how

many of these belong to the class M . In this chapter we list all matroids on sets up to size 6, and by making use of the excluded minor

property we identify those which are not in M .

It is necessary first of all to establish a method of counting

matroids on small sets. The following definition and lemmas are to

that end.

For any T = E, the restriction MIT of a matroid M on a ground

set E is a simple matroid associated with M , or a canonical matroid

of M if

T n o(1) = (f) ,IT n a(a)I = 1 for all a E E\a().

LEMMA 4.1 MIT is a simple matroid.

Proof: arest

.((1)) = n T = 4) and arest.

(a) = a(a) n T = a ,

since IT n a(a)I = 1 . //

LEMMA 4.2 All simple matroids associated with M are isomorphic, and

maximal simple restrictions of M . Any restriction of M isomorphic

to these simple matroids associated with M is itself associated with M.

Proof: Let MIT and MIT' be simple matroids associated with M.

Then there exist bijections a:a(a) -*aET, a((p) 4- (1) and

0:a(a) 4- a E T', a((p) 4- (1) , whence there also exists a bijection

-1 0a = 0:1 u a((1)) T' u

1 , r i=1

Suppose I = T and 0(I) is dependent, i.e. there exists a E T'

33.

such that a E a(0(0\a). Then a E a(U a(b):b E O(I)\a) , whence

a E a({0 -1 (b):b E 0(I)\a), and it follows that a(a) and hence 0 -1 (a)

is a member of the same closure. This contradicts the independence of

I , and we conclude that 8(0 is independent. Therefore 8 is an

isomorphism from MIT to MIT' .

For a“,aEa(b) for some bET, so MITuais not simple.

Therefore MIT is a maximal simple restriction.

Suppose MIT'=4 MIT and MIT is associated with M . Then

obviously T' does not contain two elements, one of which is in the

closure of the other, whence IT' n a(a)I = I for all a € T', and also

T' n cr(cp) = (I) . Therefore MIT' is associated with M . //

LEMMA 4.3 Two matroids M and M' are isomorphic exactly when there

is a mapping -0:E E' such that OI T is an isomorphism of associated

simple matroids MIT and WIT' and la(a)\a(cp) I = lo s (e(a))\a l (4)) I for all a E E\60)and 10()1 = 10'()1 •

Proof: M, M' isomorphic implies that there exists 6:E 4 E' , whence

8 1T is a bijection of T onto T' and e lT (I) is independent in

WIT' for I independent in MIT. Also 0 being an isomorphism

guarantees la(a)\a(cp) I = 10 1 (0(a))\0 1 ()1 and la()1 = 10 1 (01 .

Conversely suppose there exists 0:E E' such that

is an isomorphism of MIT and WIT' , and

la(a)\0((p)1 = la'(8(a))\(5 1 ()1 and la(p)1 = laW1 . Suppose 0(I)

is dependent in M' , while I is independent in M . Then there exists

J c I such that e(J) is independent in M' and a E a 1 (0(0)

for a E I\J . We take J' c T' such that J' consists exactly of

single representatives of the closures of all elements of 0(J). Then

34.

a E 0 1 (X) , whence a u 6-I(J') is dependent in M. There exists

K c I, IKI = IJ' I = IJI , such that K consists exactly of single

representatives of closures of all members of 6-1(X) and K U a

is dependent. The latter is impossible whence 6(I) is independent in

M' if I is independent in M. The conditions upon the size of the

closures of the empty set and of singletons ensure that 6 is a bijection

and hence an isomorphism. //

Every member of a set of pairwise non-isomorphic matroids on a

ground set of size 6 has a canonical simple matroid, and of course a

number have'the same canonical simple matroid. On the other hand every

simple matroid on a ground set of size up to 6 can be extended to a matroid

on a ground set of size 6 by the inclusion of additional elements in

the closure of the empty set or of one or more of the elements of the

simple matroid. Therefore the matroids on a ground set of size 6 partition

naturally into classes, each class being the non-isomorphic matroids

having the same canonical simple matroid. Lemma 4.2 says that the sameness

is only to isomorphism, i.e. the classes are distinguished by having

associated pairwise non-isomorphic simple matroids. It is easy to list

all the non-isomorphic simple-matroids up to size 6. We do this by

taking the set E = {1,2,3,4,5,6} and listing the simple matroids

MIT for some T = {r E E:r < m+1} , m = 0, 1, 2, ..., 6.

Associated with each matroid M having M' as a canonical simple

matroid we have the partition fy0 i ml , where E i is the

closure of i and E 0 is the closure of the empty set. (E 0 of course

may be empty). It follows from Lemma 4.3 that two such matroids M 1 , M2

• 2

having partitions {E} {E.} are isomorphic exactly when there exists l

an automorphism 0 of M' such that IEI IE.I , for i = 0, 1, 2, ... m.

• 3 • •

4. rank 3

35.

Therefore for each M' we count the number of partitions which pairwise

do not have this property.

Si ze, We first list the simple matroids on a set of/tat most 6. We know

that there are 43, and as all are sub-matroids of ordinary euclidean

space we so represent them. Where possible they are also shown as

graphs underneath.

TABLE 1

Simple Matroids on T = (1)

- 0. rank 0'

Simple Matroids on T = {1} •1

1. rank 1

Simple Matroids on T = {1,2}

411■ 41,

•

2. rank 2

Simple Matroids on T = {1,2,3}

3. rank 2

4

6. rank 3

8. rank 4

Simple Matroids on T = {1,2,3,4}

2 3 4 • • • •

36.

4 •

5. rank 2

4 •

/ 2 3 • • •

7. rank 3 .

Simple Matroid on T = {1,2,3,4,5}

/ 2 34 • •

3

9. rank 2

•4

10. rank 3

• • 3

11. rank 3 12. rank 3

2 3

13. rank 3

14. rank 4

17. rank 5

2 I •

So •Z 3 4 ...1" 6 • • • •

18. rank 2

•3

4 5" 6' •

20. rank 3

.4

19. rank 3

21. rank 3

15. rank 4

5 points in general

position in E4

Simple Matroids on T.= {1,2,3,4,5,6}

37.

16. rank 4

22. reok 3

23. rank 3

38.

2 3

3

4

24 rank 3

26. rank 3

28. rank 4

4

32. rank 4

25. rank 3

• 6-

3

27. rank 3

3 29. rank 4

31. rank 43

33. rank 4

plus one pt. in jr 4th dimension

6 Pts. in general posn. in E5

/ z

skew

4 5

38. rank 4

5 Pts. in general posn. in E 3 plus one pt. in 4th dimension

40. rank 5

34. rank 4

36. rank 4

39.

35. rank 4

37. rank 4

43. rank 6

6 Pts. in general posn. in E 4

39. rank 5

Abyplus one pt. in 5 4th dimension

0

Jr 6' • 41. rank 5

4 •

40.

The table below lists all the matroids on a ground set of 6

elements in terms of the simple metroids with which they are associated

and the partitions described above. The column MIT lists the simple

matroids as numbered above.

TABLE 2

ITI MIT 1E0 1,1E 1 1 ..., 'Ern i Notation Cumulative number as for M Total in Table 1

0

1 1

0 6 0.1 1

51 1.1

42 1.2

33 1.3

24 1.4

15 1.5

06 1.6 7

2 2 411 2.1

32 1 2.2

231 2.3

222 2.4

141 2.5

132 2.6

051 2.7

042 2.8

033 2.9 16

3 3111 3.1

2211 3.2

1311 3.3 1221 3.4

0222 3.5

0321 3.6

0 4 1 1 3.7

4 3111 3.8

2211 3.9

1311 3.10

1221 3.11

0 2 2 2

0321

0411

3.12

3.13

3.14 30

4 5 2 11 1 1 4.1

1 2 1 1 1 4.2

0 2 2 1 1 4.3

0 3 1 1 1 4.4

6 2 1 1 1 1 4.5

1 2 1 1 1 4.6

0 2 2 1 1 4.7

0 3 1 1 1 4.8

2 1 1 1 1 4.9

1 1 1 1 2 4.10

1 21 1 1 4.11

0 2 1 1 2 4.12

0 2 2 1 1 4.13

0 3 1 1 1 4.14

0 1 1 1 3 4.15

8 2 1 1 1 1 4.16

1 2 1 1 1 4.17

0 2 2 1 1 4.18

0 3 1 1 1 4.19 49

5 9 1 111 1 1 5.1

0 2 1 1 1 1 5.2

10 1 1 1 1 1 1 5.3

0 2 1 1 1 1 5.4

11 1 1 1 1 1 1 5.5

0 2 1 1 1 1 5.6

0 1 1 1 2 1 5.7

12 1 1 1 1 1 1 5.8

0 2 1 1 1 1 5.9

0 1 2 1 1 1 5.10

13 1 11 11 1 5.11

0 2 1 1 1 1 5.12

0 11 1 1 2 5.13

14 1 11 1 1 1 5.14

0 2 1 1 1 1 5.15

41.

15 1 1 1 1 1 1 5.16

0 2 1 1 1 1 5.17

0 1 1 1 1 2 5.18

16 1 1 1 1 1 1 5.19

0 2 1 1 1 1 5.20

0 1 1 1 2 1 5.21

17 1 1 1 1 1 1 5.22

0 2 1 1 1 1 5.23 72

6

18 6.1

to all are to

43 0 1 1 1 1 1 1 6.26 26

98

LEMMA 4.4 There are exactly 2n pairwise non-isomorphic members of

M on a ground set of size n.

Proof: Without loss of generality we can choose one ordering of the n

elements of the ground set E from all the orderings imposed by the

various functions i which induce the matroids of pol on E. Each of

the matroids of rank r on E is distinguished by the first elements of

E on which the standardised function 1 takes the values 1, 2, ..., r.

There are (n) ways of choosing those elements in the correct order,

i.e. there are ( n ) matroids of rank r on E. Summing from r = 0

to r = n we have that there are 2n

matroids on E. //

It is interesting to note that 2 n is exactly the lower bound

given by Crapo [4] for the number of matroids on a set of size n.

However a sharper bound, namely 2" /12 for sufficiently large n has

subsequently been obtained [ 3]. The sharper bound shows that pl

is a relatively small sub-class of the class of all matroids.

42.

M(T) = 1

43.

From the above lemma we see that there are 64 matroids on a

ground set of size 6 which are in M , and 34 which are not. Those

34 are distinguished by the excluded minor property of the previous

chapter. There are two possibilities for the excluded minor

kM(E

1 E2) , namely k = 2 and k = 3 . The minor given by k = 2

is a graphical matroid consisting of two rank 1 circuits, i.e. two sets

of 2-multiple edges. The k = 3 minor is the matroid whose euclidean

representation is two non-intersecting three points lines in the same

plane.

We list all 98 matroids on a set of size 6 and distinguish those

which are not members of M by an asterisk. We use the notation listed

in the above table for all of the matroids, and where possible we

represent them as graphs.

1. 1

1.2 1.3

soi•

1.4 1.5 1.6

3.7

2.4 2.5 2.6

• c•C>• EE

3.3 3.1 3.2

M(T) = 2

2.1 2.2 2.3

C>ED •

2.7 2.8 2.9

M(T) . = 3

3.4 3.5 3.6

44.

3.8

c>./"-■,\

+ 67\

3.9 3.10

3.13 3.11 3.12

M(T) = 4

45.

gc>". 3.14

M(T) = 5 These are not graphic, and are represented in euclidean space.

la(01 = 2 • la(0)1 = 1

4.1 4.2

•

•

4.3 4.4

M(T) = 6

&E1 4.5 4.6 4.7

46.

N/\

4.8

II(T) = 7

49 4 10 4.11

4.12 4.13 4.14

4.15

M(T) = 8

4.16 4.17 4.18

4.19

• • •

5.6 5.7

5.9 5.10

47.

M(T) = 9 These are not graphic and are represented in euclidean space.

= 1 • • • •

5.1 5.2

M(T) = 10 These are not graphic and are represented in euclidean space.

lo((p )I = 1 • co

•

5.3 5.4

M(T) = 11 These are not graphic and are represented in euclidean space.

• •

la(q))1 = 1 •-•-•

5.5

M(T) = 12

oc›<I> 5.8

M(T) = 13 These are not graphic and are represented in euclidean space.

• •

l o((p)1 = 1 • • • • I • • • 5 5 •

5.11 5.12 5.13

M(T) = 14

5.14

5.15

• • •

M(T) .= 15

48.

5.16 5.17 5.18

M(T) = 16

5.19 5.20 5.21

M(T) = 17

5.22 5.23

M(T) = 18 to 43

These matroids (6.1-6.26) are precisely those listed above under

the heading "Simple Matroids" on T = {1,2,3,4,5,6}, and so they are

not listed again. However those which are not in

are shown for

completeness.

6.4 (Non-graphic)

6.5 (klorf-graphic

<1> 49.

6.6 (Non-graphic) 6.7 (Graphic)

6.9 (Non-graphic) 6.14 (Non-graphic)

6.15 (Graphic) 6.18 (Graphic)

6.20 (Graphic) 6.21 (Graphic)

As can be seen from the above, there are 68 graphic matroids on

a set of size 6, 42 of which are in M and 26 are not. There are

Sø non-graphic matroids, 22 of which are in M and 8 are not.

The calculation of the number of non-isomorphic matroids on a set

of 6 elements seems to be a new result, and so we state it as a theorem.

THEOREM 4.5 There are 98 non-isomorphic matroids on a set of 6 elements. //

CHAPTER 5

In the previous chapter we saw that not all matroids of the class

M are graphic. It is natural to enquire whether they are a subclass

of any well known class of matroids, and in this short chapter we

answer the question as well as establishing a necessary and sufficient

condition for a matroid - to belong to M.

LEMMA 5.1 A matroid M E M on a ground set E is a transversal

matroid.

Proof: We construct the family of subsets U = (A 1 :1 i p) ,

where A. = {a E E:1(a) By Lemma 2.10 a transversal of U

is precisely a basis of M , whence partial transversals are precisely

the independent sets of M . //

This is a most interesting result because M* being also in M

is also transversal. Therefore here we have a subclass of transversal

matroids whose dual is also transversal. That not all transversal

matroids have duals which are also transversal is shown by the following

example. Figure 4 below is a graphic matroid which is transversal, and

its dual (Figures) is also graphic but is - not transversal.

50.

Fig. 4 Graphic Matroid

Fig. 5 Graphic'Matroid

which is transversal

which is not transversal

51.

The above are the matroids 6.15 and 3.5 of the previous chapter

and of course they are not in M.

Not all transversal matroids whose duals are also transversal,

belong to M. The following example shows this.

%.3

Fig. 6 Graphic Matroid

transversal but jV

Fig. 7 Transversal Matroid

dual of Fig. 6

The matroid of Fig. 6 is transversal with family

(11,21, {3,4}, {1,4,5}) , and the family of the transversal matroid of

Fig. 7 is ({1,2,5}, {3,4,5}) . They are obviously dual. The

excluded minor characterisation shows that they are not in M .

The following theorem shows precisely which transversal matroids

are members of

THEOREM 5.2. A matroid M on a ground set E is a member of M if and only

if it is transversal having a presentation of a family of nested sets.

Proof: Given M(E) E M we construct a family U = (A i :1 i 5- ,

where IE. = fa E E: 1(a) ?. i} , and p is the rank of M(E) . The

E. form a chain ordered by strict inclusion.

Conversely let M(E) be a transversal matroid with family of

representablesets[J--(E.11 i p) such that El

D E2.. . D Ep

.

52.

For all a E E 1 \E i+1 , 1 5 i 5 p-1 , we assign 1(a) = i , and

for all "a E Ep we assign 1(a) = p . The function 1 induces a

matroid of the class M on E and the transversals of U are

precisely the bases of the induced matroid.

//

We conclude this chapter with a necessary and sufficient condition

for a matroid to belong to the class M

THEOREM 5.3 Let M(E) be a matroid of the class M whose independent

sets are the family 1 . Let C be the family of circuits of M(E) .

Let

(I:IE J,iIwherebisacoloop) 1

and for I

C I = (C a : a E C a E C, a E I, IC a lmin)•

(4 particular Ca might not be unique, and the family might have

some :repetitions of Ca 's.) Then M(E) E M if and only if for

any C I , at most i circuits have cardinality 5 i+1 for 1 i 5

Proof: Suppose M(E) E M and E is p-ordered. Consider any

I = {ajl , aim} E and let C I = (C 1 , •.., Cm ) , where

C. n a t (1) and IC .I is minimum. Then IC.I > p a ji ( ) i ji

from Lemma43and at most i circuits have cardinality 5 1+1 .

This applies for 1 5 i 5 II1-1.

Conversely let M(E) be a matroid with circuit structure as

described. We define a function p on the set E as follows:

if a is a loop, let p(a) = 0 ;

if a is a coloop, let p(a) '= p(M)

if a is neither a loop nor a coloop, let p(a) = ICal - 1 ,

where Ca n a cp and IC a I minimum.

53.

This function i induces a matroid Mi(E) E M , with the family of

independent sets J . We have to show that J = J . For any I E I

let I = {b1' "• , b5 } U b5+1 , bt} , where the first subset is

a member of I' and the second is not. Because of the assumed

circuit structure, I E .

For J E we let J = {c 1 , ..., cm } where i > j=> p(c)

Suppose for some i < c i } is independent in M(E) but

{c 1' c i+1 } is dependent. Then the latter contains a circuit of

size at most i+1 and that circuit must meet c i+1 . Therefore

1.1(c i+1 ) = i , which is impossible since J is independent in Ms (E) ,

so we conclude that the independence of {c 1' • .., c.} implies the

independence of {c 1 , c i+1 } . Since at most one of the circuits

of M(E) meeting {c 1 , cd has cardinality 5 2, {c 1 , cd is

independent in M(E) , and induction on i gives us that J is

independent in M(E). //

54.

CHAPTER 6

In the previous chapters the ground set upon which the matroid is

induced is finite. This chapter deals with infinite ground sets, and in this

case we use the term pregeometry rather than matroid. From Crap and

Rota 5] we have the following definition:

A pregeometry G(S) is a set S endowed with a closure relation a

having the following properties:

(i) the exchange property: if a E o(A U b) and a a(A) , then

b E a(A U a),

(ii) the finite basis property: any A = S has a finite subset

Af E. A such that a(A

f) = a(A).

(We recall that a closure relation a is defined by the properties

(a) A c a(A) , and (b) A ca(B) e0a(A) c a(B) ,

for all A,B c S . )

As for matroids, a pregeometry has a family of independent Sets, and

the pregeometry is completely defined by this family. The family I

turns out to have the same properties as the collection of independent

sets of a matroid, namely:

(1) J=IEI J I

(2) I,J E I and III > NI there exists an element x E I\J

such that JUXE I.

In addition property (ii) above, the finite basis property, ensures that

(3) all members of I are finite, and have finitely

bounded size.

Therefore a pregeometry G(S) consists of the non-empty set S , together

with a non-empty family I of subsets (called the independent sets) of

satisfying (I), (2) and (3) above.

55.

We are concerned in this chapter to show that the characterisation

of the class M matroids revealed in Chapter 3 carries over to the

class of pregeometries induced in the same manner as M .

A restriction GT(S) of a pregeometry G(S) , or a subgeometry as

Crapo and Rota call it, is the set T endowed with the closure relation

arest given by

a t (A) = a(A) n T.

It is easy to show that a restriction is a pregeometry.

LEMMA 6.1 If G(S) is a pregeometry on a ground set S and T is a

finite subset of S , then G1 (S) is a matroid. //

LEMMA 6.2 Let the function p:S -0- Z be bounded above. Then p defines

a pregeometry G(S) whose family I of independent sets is given by

E 1, and I E 1 if and only if max{p(a): a E J} IJI

for a// (P#JcI.

Proof. Since p is bounded above we are assured of the existence

of max(p(A): a E A} for all A c S. Otherwise the reasoning is the

same as in Lemmas 2.1 and 2.2. //

We can, without loss of generality, assume that 0 p(a) p(G)

for all a E S , since the function v:S Z given by

v(a) = min fmax{0,p(a)}, p(G)} defines the same pregeometry as G. This

assumption is made for the rest of the chapter.

We call pregeometries derived in the above manner pregeometries of

the class G .

The next proof requires Rado's Selection Principle which is as follows:

56.

Let U = ( A i : i E I) be a family of finite subsets of a set S. Let

J denote the collection of all finite subsets of the index set I and

for each J E j , let 0 be a choice function of the subfamily

(A.: i E . Then there exists a choice function 0 of U with

the property that, for each JEj, there isa K with• JcKEj

and 0 O KIJ . (For proof see Mirsky

THEOREM 6.3 The pregeometry G(S) is in 6 exactly when each of

its finite restrictions (submatroids) is in m

Proof: If Gp €.G then M = Gp IT is defined by pi T , using the

same reasoning as in Lemma 2.23.

Conversely if for each T cc S , Gil = M for some pT :T Z , PT

we define a family (X) s by

Xa

= {0, 1, 2, ..., p(G)} for all a E S.

Then for each T cc S the function p T is a choice function. Rado's

Selection Principle ensures the existence of a choice function

p:S Z with T c K cc S p IT ='KIT = TIT , and as Xa ()CO}

for all a E S , this choice function is bounded. The function ji

induces a pregeometry Gp on Z.

It remains to show that G is identical to G. This will be so

if I independent in G il <=> I independent in G .

If I is independent in G then max{p(a): a E J} IJI for all

c I , whence max{pIK

(a): a E J} IJI for I c K cc S. Therefore

I is independent in M = GIK , and hence in G . lK

Conversely if I is independent in G then I <co and there

exists K with I c K cc S such that I is independent in GIK =M PIK

57.

• Therefore max{p iK (a): a E J} IJI for all J = I , whence

max{u(a): a E J} IJI for all J c I , and I is independent in G. //

We define a minor of a pregeometry to be any contraction of a

finite restriction.

THEOREM 6.4 G is characterised by the family kM , k = 2, 3, ...

excluded minors.

Proof: This follows immediately from Theorem 6.3 and Theorem 3.9. //

THEOREM 6.5 Each G c G is characterised by having a finite chain P

o() .= F0 c FI --

k s S,. where F11 is the unique minimal

1

-

non-trivial extension of the flat 1 , unless F has no. such extension . _ . ._ . i .

in which case F i4.1 = E. Each flat in Go is a direct sum of some F.

and a free matroid.

Proof: If a pregeometry G has such a chain then so doesany finite

restriction. Therefore, by Lemma 3.3., any finite restriction is a

matroid in m , whence G E G.

Conversely let G E G and suppose G has two minimal non-trivial

extensions F and F' of a flat H. Then there exist circuits C c F

and C' c F' with • f(C) = f(F) and f(C 1 ) = f(P) . Now in the

restriction GIC U C',Grest(F)

and arest(F') are minimal non-trivial

extensions of arest (H) , but since the restriction is in m we have

arest

(F) = arest (F') , i.e. arest (C) = arest(C')• It follows from

this that f(C) = f(C 1 ) , whence F = F' . We begin the chain with the

closure of the empty set and from the above the rest follows.

Any flat is (i) free, or (ii) an F i , or (iii) a free extension

of an F. . Therefore a flat F is the direct sum of F. , for some

0 i k , and the free matroid MI(F\F i) • //

58.

THEOREM 6.6 If G E G then it is transversal.

Proof:LetS...{a E S: p(a) i} for i = 1, 2, ..., p(0). Then

U = ( S.: I 5 i p(G)) is a family of subsets of S whose transversals

are bases of G . //

THEOREM 6.7 G if and only if it istransversal having a

presentation of a family of nested sets.

Proof: Suppose G E G . Then by Theorem 6.6 it is transversal and its

family of representable sets has the desired property.

Conversely let G be a transversal pregeometry with family of

subsets U = ( S i : I 5 i p(G)) having the property S p c s p _ l ... S l .

We define a function p:S Z by

p(a) = i if a E Si \S i+, for I 5 i p-1 ,

(a) =p if a E S

p(a) = 0 if a

Then the pregeometry G induced by p has as its bases sets which

can be described by B = fa y ap: p(a i ) i for 1 i p} .

It is obvious that the transversals of U and the bases of G are

precisely the same, i.e. G = Gp E G //

The other properties of the class M carry over to the class G

where appropriate.

59.

CHAPTER 7

In this chapter we examine the matroids induced by p:E r Z via

the submodular function f:2E 4- Z. This somewhat enlarges the class M ;

for instance some simple graphical matroids were excluded from ri but

are induced by the function defined on r-sized subsets. It also

provides matroids with a richer structure.

We begin with the function p:E r Z and, as in Chapter 2, obtain

f:2 Z as follows. Let

f(A) = max{p(a l ,...,a r ): a l ,...,a r EA} for all (ptAcE,

f(A) = min{p(a l ,...,a r ): a l ,...,a r E E} for A =

Functions derived in this manner are said to belong to the

class Fr .

Functions of this class are always increasing functions, but they

are not always submodular. Consider for example p:E2

Z defined

as follows. Let p(a,b) be the integer part of the distance between

the points a and b in Euclidean space E . Let A = fa,b1 ,

B = {c,d}, and p(a,b) = 5 , p(c,d) = 5, p(a,c) = 12, p(b,d) = 12.

Then obviously f(A) + f(B) < f(A U B) f(A n B).

There are some functions on r-sized subsets which induce submodular

functions in the manner of Chapter 2 but it is not the purpose of

this thesis to characterise them, if indeed this is possible. However

we construct one such function as follows:

Let p i , pr be functions from E into Z . We define a function

p:Er Z by

..,a r ) = + p (a 'for all a E E. r r i

60.

It follows that f(A) = max{p1(a

1) + + pr (a

r): a

1"'"ar E Al ,

whence f(A) = f 1 (A) + + fr (A) , for all A c E , where

f 1 (A) = max {p i.(a): a E A) •for 1 i r. In order to distinguish the

function f from that of Chapter 2, we designate it f r and we have

fr = f + + f

r 1

LEMMA 7.1 The function fr:2 E Z is submodular.

Proof. For any A,B c E

f r (A u B) + fr (A n B) = f l (A u B) + + fr (A u B) + f l (A n B) + • •

+ fr (A n B)

f l (A) + f l (B) + + r (A) + fr (B)

= fr (A) fr (B) / /

The function p therefore defines a matroid on E . It is natural

to enquire what functions on r-tuples are expressible as sums of

functions on singletons. The anSwer is that there are not very many,

as the next lemma shows.

LEMMA 7.2 A function p:E x E 4- Z can be written

m(ap ai ) = Pi(a) + p2 (a i ) for all av a i, E E if and only if

( pah) - P(apa k ) = P(apah) - P(ap a k ) for all ap ap a n ,a k E E.

Proof. If p(a.,ej.) = p 1 (a i

) + 2 (a) for all a.,aj then by

substitution we have p(apa h ) - P(apa k ) = P(ap a h ) - p(a j o k ).

Conversely suppose we have a function p:E x E Z such that

p(a h ,a k ) - u(a h ,a i ) = p(ai,a k ) - p(a p aj ) Then , p(a.,a .) is 3

determined by the 2n - 1 terms p(a v a l ) p(ar a n ),p(a 2 ,a

p(a n ,ai), where 1E1 = - n. We must show that there exist

p 1 ,p2 :E Z such that the n2

equations

61. "

p1(a 1 ) + p2 (a j

) = p(a.,a.) are satisfied. There is an infinite 1 J

number of solutions for p 1 (a 1 ), p i (a n ) , p 2 (a 1 ), ...,p 2 (a n ) to

the 2n - 1 equations

p1(a

1 + p

2(a ) = p(a

l'a )

p (a l ) + p2 (a n ) = p(ar a n )

p i (a 2 ) + p2 (a l ) = p(a 2 ,a 1 )

= p(a n ,a 1 ) ,

and by fixing an integer value of say p i (a l ) we obtain one integer

value for each of the others. It remains to show that this solution

is consistent with the remaining n2

- (2n-1) equations. This is so

since

p1(a

i) + p

a. 11 (a l ) 11 1 (a 1 ) 1-12 (ai )

11 1 (a 1 ) 112 (a l )

= p(a tp l ) + p(ar a i ) - p(a 10 1 )

p(a.,a.) j

Similar but more complicated results apply for p defined on

larger subsets. '

LEMMA 7.3 deleted

/ /

62.

Since fr is submodular it defines a matroid on E whose

independent sets are given by 1 = {I: f r (J) IJI for all J c I} .

We designate this matroid as M r , and say that it belongs to the

class Mr .

We define the union of r matroids M1'

, Mr

on E as the

matroid whose independent sets are each precisely the union of r

subsets of E , each of which is independent in a distinct M i . The

matroid M1 u u M

r is defined by the collection 1 of.independent

sets given by I = {I: I = u u I r , E , where is

the collection of independent sets of the matroid M i .

LEMMA 7.4 Suppose f2

= f l + f2 and that f2,f 1 and f2 induce the

2 ' matroids M, M1 and M2 respectively on a ground set E . Then

2 M = M

1 u M

2

Proof. For any 1 1 ,1 2 independent in M 1 ,M2 respectively,

e2 T T C (T T T I (1 1 U 1 2 / = 1 1 %1 1 U 1 2 ) ' 1 2 (1 1 U

T 1 2 ) f 1 (1 1 ) f2 (1 2 )

11 1 1 + 11 2 1 ?. Il l u 1 2 1.

Conversely suppose •there exists a set I of cardinality m+1 which

is independent in M2

but cannot be partitioned into two sets, one

of which is independent M 1 and one in M2 . Further suppose that

all sets of size can be so partitioned. We choose a E I such

that m l (a) m 1 (x) for all x E I\a , and partition I\a into

63.

I and I2

which are independent in M and M2

respectively. 1 1

Then f 1 (I) = f 1 (I 1 ) = 11 1 1 = p i (a) , and furthermore there exists no

b EI such that p 2 (b) > f 2 (I 2 ) , otherwise I could be partitioned

as required.

The set 1 2 U a is dependent, whence f 2 ( J 2 U a) = 1J 2 1 for A

some J 2 = Iv Suppose J 2 = 1 2 ; then f2(I) = f 1 (I) + f 2 (I) = II 1 1

+ 11 2 1 < III , which contradicts the independence of I in M2

.

Therefore J 2 = 1 2 and p 2 (a) f 2 (J 2 ) = 1J 2 1. If there exists no

b E I such that p 2 (b) > f2 (J 2 ) then f(I i u J 2 U a) = II 1 1 + 1J 2 1 ,

which is impossible since the set I U J 2 U a is independent in M2 .

Such an element therefore must exist, and by interchanging a and b

we obtain I and gi) U b. Again there exists c Ub

AO ) and r r (T HM\J (i) such that f (3 (i) J (i) (r) with J 2 = _ 2 ___" ) = 1_ 2 1 = ( otherwise I would partition as required. But then (i) Ai J 2 i) U c

would be dependent in M2

, unless it is possible again to interchange

elements as above. The latter must be true, and in this manner after a

(s) finite number s of interchanges we arrive at I , I

(s) of the

2

same size of I1

and 12 respectively, and an element x not an

element of either, such that f1 (I) = f 1 (I) = II 1 and

1 1

f 2 (I) = f 2(45) ) = 11 2 1 = p2 (x) , which contradicts the independence of

I in M2 . We conclude that if independent sets of size m in M2

partition as required, then so do those of size m+1 .

For III = 1, f 2(I) _ 1 implies that f i (I) 1 for

= 1 or 2

orboth,whenceI=I 1 U1 2 whereLis independent in M. for

i = 1, 2. //

We now extend the lemma to the general case of the union of

r matroids.

64.

THEOREM 7.5 Suppose fl , fr E F and define matroids

Mr E M respectively, and that fr = f l + + fr

defines the matroid Mr E Mr . Then Mr = M 1 u uM .

Proof. Suppose Mm = M1 u U Mm for some m < r, and there exists

I independent in e l which cannot be expressed as the union of

m+1 sets, each independent in a distinct M i . We take the union of

maximal sets of I , each of which is independent in a distinct M i ;

this is obviously a proper subset of I . Therefore there exists

a E I which when joined to each of these maximal sets forms a set

which contains a circuit in the appropriate M i .

\ It follows from the above that _ _ is dependent in Mm : Using

this and the fact that I is independent in M m+1 , we have

im4- 1 (J) fl (J) .. . fm+1 (J) for 'a ll•

but fm(K) = f l (K) + + fm (K) < 1K1

for some K c J . If K i is the subset of K . which is independent in

M. , we have f. (K) f.(K.) .?. 11(.1 , whence e l (K) IKI - 1 . But i 1 1 1 i

e+1 (K) IKI , so it follows that fm+1 (K) 1. Therefore there exists

b E K such that i (b) 1. Suppose b = a ; then a is independent

in Min+, , which is contrary to our original supposition. Suppose

b a ; then b € I. for some i and p i (b).# 0. It follows that

(I0{b}) u {a} is independent in M i and b is independent in M mil. ,

which also contradicts our original supposition. Therefore

Mm

= M1 u u M implies that Mm+1

= M 1 u . Lemma 7.4

shows that the result is true for m = 2 , so it is true for m = r by

induction. //

LEMMA 7.6 deleted

65.

LEMMA 7.7 Suppose p i (a) = 0 or 1 for all a E E and for

i = 1, r . Then a set {a 1 ,...,ar} is independent in Mr (E)

if and only if the p i 's can be permuted such that p. (a.)1 for

Proof. Clearly the set is independent if such a permutation does exist.

Conversely assume {a l , ..., a r} is independent. Suppose the

result is true for independent sets of size m < r , i.e. there exist

such that p(a1) = 1 for i = 1, 2, ..., m . If it is

not true for m+1 then pj(am+1

. ) = 0 for j > m Also if ii(a) = 1 J 1

for j > m and i < m then p.(am+1 ) = 0 , otherwise by rearrangement

of the p's the theorem is true for m+1 . But since am+1

is

independent some pi .maps it to 1 , so we conclude that p. 1(a.) = 0

J

for j > m and i < m. This plus pj(am+1

) = 0 for j > m gives us•

that f({a l ,...,am+1 }) = m, which is impossible since the set is

independent. Therefore if the result is true for •m it is true for

m+1. Clearly the result is true for m=1. //

Suppose now that p(a l ,...,a r ) ='11 1 (a 1 ) + + pr (a r ) and that

the maximum value of any p i on E is k . Then we can write

ik ' where

if j lc .(a)

and p..(a) = 0 if j > 1J

In our usual way we define a function fij

:2E

Z(2) as follows:

f 1 .(A) = max{p..(a):a E A}. J 1J

Then for any A c E it is easily verified that

f 1 (A) = fl(A) + + f ik (A). Therefore f = fll f ik

LEMMA 7.8

by mu .

= M11

u u Mrk •

whereM i is the matroid defined j

Proof. It is only necessary to prove that M i = Mil u u Mik .

If I is independent in M i then f i (J) IJI for all

J c III , whence there exist distinct am

E M = 1,2,..., III,

such that p i (am ) m, i.e. p im (am) = 1. Therefore am is

independent in M im and M i c Mil u u Mik .

Conversely consider a union I n u u I ik , where I im is

independent in M im . We suppose that all the I im are non-empty

since if the inclusion we seek is true for this, it is true for some

empty. Now I im = {a}, where a is such that 1i(a) = 1 for j m,

which implies that p i (a) m. Therefore' ii u u I. is independent

in 'M. , whence M. c M. . Mil u

u ik — //

We designate the closure in the matroid M r E Mr by the relation

ar:2

E 2E. The following explores the structure of closures in the

class Mr and their relation with closures in matroids of the class

1.

LEMMA 7.9. In the ,matroid Mr (E) E Mr , for all A c E,

ar (A) = A u B „where B is maximal with respect to fr ( ) = IJI

66.

L T =1 (V)ap uazj W 3 p:waqvw

azj u dyi9uo-14v7aa aansolo ayq ss -4- z: 1-1) j± *II'L VWW31 • 3

I I

III = ! a il + + I I II = ( I).1 os pue '3 'T =

aoj 1 . '1 = (W.1 ueia 'sapaadoad paapbaa aq4 6uLiteq

L L

• I 1414m I 0 0 I = i asoddns SL8saanuo3

• •w u .uapuedepu!. jo 4asqns LewLxew e s!. • os pUP

L L OI = ( W.1 Jall4a1J '1I1 = (* I)*.J Pue a I 0 0 I I = I

aouaqm 'I a il + + z ( aIr1.1 + + ( T I) TJ III

aaojaaaql 1 111 z OW4 z ().4. 4E144 Pue !•14 u!. 4uapuadapuL s!.

.1 I aaalm I n '''n I = I 4E444 mom' am 'III = () 1 asoddns •400.1 d

'z 'T = I Jo; . (Vj puv 'W 3 1 14 U2 I fo qasqns

quapuadapu:t pourtxvw v 1. 1 'A l 0 I I = i .4 R2720 pup f:1,

III = * aW D J 14 tri, quapuadaptel, s-.1. asoddns '0I 'L WW31

'4as

5Lq4 4noqe aaow sn sija4 emal. 4x8u

aqi *IN = (r),J 4eq4 Lions p .es 4uapuadapuL alp 4noqe pauaamoo

aae am aansop aq4 jo aan4ona4s aq4 6uLuLwaa4ap u snoLAqo

/ / .9r1v. (v) ap aaojaaaqi •anoDenv pu ae

aouaqm (N) aj (p) aj uaqi (N) aj (!4) pue 4uapuadapuL

s!..) (L) . o4 4oadsaa q4Lm LewLxew s!. j N aaaqm INI (0) aj

o4 4oadsaa 44Lm LewLxrwes!. a aaaqm 'anoDv asoddns

.E1 Ri. snopqo •anoqe

paqposap se n v 4as e aneq am 3 v kue J04 asoddns . jooad

= ( 0 ).A J (4) Puy (3) aw

quapuadapu?, p oq qoadsaa 1.01,m 7vur2zvw sl V D p aaaym

'L9

68.

Proof. Suppose D c ar (A) . We let D = D i D2 with D 1 c A\B

and 02 c B\A, where B is the set defined in Lemma 7.9. Obviously

D C U a.(A) . If J = J U ... 0 J is the maximal independent 1 — 1 r

set contained in A such that f r (J) = IJI , then J i C I i for

i = 1, 2, ..., r , where I i is the maximal independent set contained

in A such that f(I) = 1I . Therefore fr (D2 ) 1J 1 1 + + 1J r 1

II 1 1 + + li r , whence f(D 2 ) Il i ' for at least one i , and

so D 2 E. o i (A). //

We move on now to consideration of circuits in matroids of the

class Mr

LEMMA 7.12. If C is a circuit in the matroid Mr E Mr then

C C U U Cr , where C. is a circuit in Mi E M .

Proof. If C isacircuit in Mr then for any aEC,C=Jua

where J is independent in Mr and f(C) = f(J) = 1J1. Therefore

by Lemma 7.10 J = J I U ... 0 J r , J. being a maximal subset of J

independent in M i , and p i (a) 1J i l for i = 1, 2, r . It

followsthenthatC.cJ.uawhereC.is a circuit in M. , and —

so C = C1 u C

r . —

LEMMA 7.13. If C is a circuit in M then for each i , 1 i r,

there exist at least two elements b,c E C such that

p i (b) = = JI.

Proof. If not then we only have say p i (b) =IJI and f(C\b)< ICI -1,

whence C is not a circuit. S //

It is now possible to construct circuits in M r . We select

independent sets J l , J 2 , ... J r from M 1 , M2 , ..., Mr respectively

1 I 11 a 3 a4 a 2

a 3 a l a 2

69.

suchthatf oralli,k,f.(J1=0.for all 11 ik

and the J1 1 s are disjoint. If there exist, for each i , at least

two elements b,c E J i U ... 0 J r such that p i (b) = p i (c) =

then we join to the above disjoint union any a E E such that 0 <

p.(a) IJ.I for all i , and this gives a circuit in M r . If

for any i , only one element b in J 0 J r is such that 1

p i (b) = IJ i l and otherwise pk(a) IJ k 1 , and this joined to the

disjoint union of independent sets is a circuit in Mr . If

J = 0 ... 0 J r a s above and there exists a E E\J such that

=IJ.Ifor all i then C=Jua isacircuit in M r and

we have C = C 1 u u Cr , where C i = J i u a is a circuit in M i .

If B1 , are the collections of bases of the matroids

Mr E M then clearly the bases of M r are the maximal

members of the family (B 1 u U Br : B i E M i ).

We turn now to the consideration of dual matroids of those in

"r . The matroid Mr* has as its bases the sets which are the

complements of maximal members of the family (B 1 u U Br: B. E

More succinctly, the bases are the minimal members of the family * * *

(B i n...nBr :B i E.), where Bi is the set of bases of M. . B,

Hence in general Mr* does not belong to Mr . However below is an

example of a member of 112 whose dual also belongs to M , in

fact it is self dual.

Let E = {a 1 , a 2' a3' 4 } with 1 l' 1 2 :E Z two standardised

functions (levels) as shown in figure 8 below.

Fig 8. 1 1 and 1 2 on E

a2 a

4 a5

1.1 1 1 1 0 0 0

2 1 0 1 0 1

p3 0 0 0 1 1

al

70.

Then land 12

define two matroids M and M

2 having collections

1 1

of bases ({a 3 }, {a4 }) and ({a 1 }, {a 2 }) respectively. The collection

of bases of M2

= M i u M2 is ({a l , a 3 }, {a l , a4 }, {a 2 , a 3 }, {a 2 , a4 1).

The matroids M and M2

have standardised functions or levels as 1

shown in figure 9.

1 1

12

3 a l 3 a4

2 a2

2 a3

1 a4

a3

1. a1

a2

Fig 9. 1 1 and 1 2 on E

The matroids M and M2

have collections of bases 1

({a4 ,a 2 ,a 1 1, {ar a , 2a 1 })

and (fa1 ,a

3 ,a4 }, {a 2 ,a

3 ,a

4 }) respectively,

2* whence the collection of bases of M s precisely that of M

2 i

.

As we remarked in the beginning of this chapter, some very simple

graphical matroids, such as that on a quadrilateral with one diagonal,

are not in M . However the class Mr

, being more complex, does

admit some of these, including the example mentioned above. This is

shown below, and we chose matroids of rank 1 to build the required

matroid.

Fig 10. Quadrilateral with diagonal

Afurthercorplicationisadrilissiblein11 5 ,wherethe are

of rank 1, as shown in figure 11.

71.

1 2 3 . 4 5 6 7 8 9 1 1 0 0 0 ,0 0 0 0

P2 1 0 1 0 1 0 0 00

P3 0 0 0 1 1 0 1 0 0

P4 0 0 0 0 0 1 1 0 1

11 5 00 0 0 0 0 0 1 1

Fig 11. M E M is graphical matroid

It seems that the graphical matroid on a chain of triangles in

the manner of figure 11 above could be represented as matroids belonging

•to the class Mr for some r. However the limitation of this class

for representation of graphical matroids becomes obvious when we

consider a quadrilateral with two diagonals, as the following lemma

shows.

LEMMA 7.14. It is not possible to find p i ,...,ur such that they

define a matroid Mr E Mr on the edges of the graph below which is

identical to the graphical matroid.

a

o'

Proof. It is possible to find the required functions if and only if

it is possible to find a certain number of functions which map the

edges to 0 or 1 only, such that these functions define the necessary

matroid. We consider then only functions mapping the edges onto 0 or 1.

Suppose it is possible to find p i , p2 , ..., i.e. f i , f2 ,

mapping only to 0 and 1, such that they define the required matroid.

72.

Then for {a,b,e} there exist, without loss of generality, f l , f 2

which each map the set to 1. Further, f 1 ({a,b,e}) = 0 for

i 1 or 2 , whence f1 (x) = f

2(x) = 1 for at least one x E {a,b,e}.

Assume x = e ; then f k({c,d,e}) = f.({c,d,e}) = 1 for some k,j,

and f i ({c,d,e}) = 0 for i t k,j . It follows then that k and j

are 1 and 2, whence f({a,b,c,d}) = 2 which is impossible. Therefore

we can assume, without loss of generality, that f1 (a) = f

2(a) = 1.

If f i ({d,f}) = 1 for i t 1 or 2 then f({a,d,f}) 3 which is

impossible. Therefore f({a,b,d,f}) = 2 which is also impossible and

we conclude that it is impossible to find the required f i , f2 , . //

We saw that matroids of the class M are transversal. This

result is now extended to the class Mr , and strengthened.

THEOREM 7.15. Mr E Mr is transversal.

Proof. From Lemma 5.1 M i , ..., Mr are transversal matroids, whence

there exist families (X) 1 (X) T of sets of E such that '1 A r

the partial transversals of (X) T for i = 1, 2, ..., r are

precisely the independent sets of M i , ..., Mr respectively.

Therefore the independent sets of M r are precisely the partial

transversals of (X) where I = I1

u U Ir

. / /

The following theorem shows that the reverse is also true.

THEOREM 7.16. Let M(E) be a transversal matroid of rank r. Then

M(E) E Mr .

Proof. Let U be a presentation of M(E) and let (E l , ..., Er )

be a subfamily of U such that its transversals are bases of M(E).

We define functions p i , ..., pr :E Z as follows. Let p i (a) = 1 for

73.

a E E. and p 1 (a) = 0 for a E.

Suppose I = {a l ,...,am} is independent in M(E). Then there

exists a subfamily (E il ,...,E im) of U with aj

E Eij

and

J J = 1 for 1 j m . Therefore I is independent in the

matroid M r defined by 1.1 1 ,...,1' r .

Conversely suppose I = is independent in M r defined

by p i ,...,pr . .Then there exist such that

1 for 1j m, where a.j:

. E E.i

and it follows that I j

is independent in M(E). //

We now define Mf

to be the class of matroids consisting precisely

of all subclasses Mr , r finite. Then we have the following exact

description of the matroids of this thesis.

THEOREM 7.17. The class of all finite transversal matroids is

exactly Mf . I/

74.

A FAILED CONJECTURE

We remarked in the introduction that one motivation for this

study was the hope of building up from functions defined on singletons

to functions defined on subsets of size r, in order thereby to obtain

a function f:2E

Z which is identical to a well known submodular

function or perhaps even to the rank function of a well known matroid.

Another approach is to begin with a submodular function f:2E

Z

and define f r :Er Z by. fr (A) = max{f(B): B c A, 1BI r} . We

know that f l is submodular and we conjecture that if, f r is

submodular then fr+1 is also. From this we would have submodular

functions f1, f

2, to f P , where p is the rank of the matroid defined

by f, and furthermore fP and f define the same matroid.

However fr being submodular does not imply that

submodular as the following example shows.

Let E = {a,b,c,d} with f:2 E Z given by

f(a) = 2 , f(b) = f(c) = f(d) = 1

f(ab) = f(b ) = f(cd) = f(ac) = f(bd) = 2 ,

f(ad) = 3

f(abc) = 2, f(bcd) = f(cda) = f -(dab) = 3

f(abcd) = .

r+1 is

Then f is increasing and submodular. We know from Chapter 2 that

f l is submodular, but f 2 is not, as can be seen by considering the

sets A = {a,b,c} and B = {b,c,d} . Then f 2 (A) + f2 (B) = 2 + 2 ,

while f2(A n B) + f2 (A u B) = 2 + 3.

75.

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INDEX

77.

Automorphism, 15

Base, 1

Circuit, 3

Closure, 2

Cobase, 3

Cocircuit, 3

Corank, 3

Dual matroid, 18

Euclidean representation, 4

Flat, 2

free extension of, 25

non-trivial, 25

non-trivial extension of, 25

Function„7

class F, 7

class Fr , 59

increasing, 4

normalised, 14

standardised, 11

submodular, 4

Graph, 2

simple, 6

Hyperplane, 3

Independent set, 1, 5

Level, 11

Matroid, 1, 2, 3

canonical, 32

class M , 7

class Mf , 73

class Mr , 62

graphic, 4

number of, 42, 43, 49

simple, 6

transversal, 4

Minor of a pregeometry, 57

p-Ordering, 10

Pregeometry, 54

class G , 55

Rado's Selection Principle, 55

Rank, 2

Restriction of pregeometry, 55

Union of matroids, 62

78.